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The Gambler's fallacy:
I fall for this fallacy when playing battle for Wesnoth - if my unit misses several times, I assume that it will hit more frequently the next few times. https://social.systemreboot.net/attachment/70815
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That's not incorrect. If your unit misses several times, then you should expect to hit more frequently after a run of misses, assuming your normal average of hits is greater than zero.
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Lets take an example. Suppose if my spearman has a 60% chance to hit an horseman. There's a 6.4% probability to miss in all three turns and 21.6% to hit in all three turns. Suppose my spearman misses all three turns, then how does the probability to hit in all three turns increase?
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* replace turn by attempt. There's 3 attempts per turn.
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If you've got a 6.4% probability to miss in all three turns, then for the next three turns I would expect a 93% probability of getting at least one hit -- much higher than the probability of missing three turns in a row again!
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93.6% in fact. I'm having a 50% probability of errors while mathing today.
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@solariiknight said "hit more frequently the next few times". We are interpreting this phrase differently. Hence, the apparent confusion.
You are comparing the probability of hitting AT LEAST ONCE to the probability of missing THREE times in a row.
@solariiknight and I are stating that the probability to hit in the next attempt is still 60%, and has not been affected by the result of the previous attempts.
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> FOR THE NEXT THREE TURNS I would expect a 93.6% probability of getting at least one hit
The probability of getting at least one hit is 93.6% for ANY three attempts, regardless of whether those attempts are consecutive, come before/after some other event, etc. Each attack attempt is completely independent and uncorrelated with the result of any other attack attempt.
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Say, you have a coin (assume unbiased) in your pocket, and you toss it. The probability of getting a heads is 50% regardless of that coin's tossing history with a previous owner. Your current toss is independent of all previous tosses.
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Exactly! So if your typical hit rate is 50%, but you have a run of misses, then for the next run I would expect a hit rate of 50% -- much higher than your previous run of misses! :-)